Question

The harmonic conjugate of the point $R(2,4)$ with respect to the points $P(2,2)$ and $Q(2,5)$ is

• A. $(2,4)$
• B. $(2,8)$
• C. $(4,2)$
• D. $(4,8)$

Answer 1

The correct option is B $(2,8)$

Given : $P(2,2)$ and $Q(2,5)$
Let $R$ divides the $PQ$ in the ratio $k:1$ internally. Then
$(2,4)=(\frac{2k+2}{k+1},\frac{5k+2}{k+1})\Rightarrow k=2$

Let $C$ be the harmonic conjugate of $R$.
Then $C$ divides $PQ$ in ratio of $2:1$ externally.
So, $C=(\frac{2\times 2-1\times 2}{2-1},\frac{2\times 5-1\times 2}{2-1})$
$\therefore C=(2,8)$